In the OP Application, the needed current as understood is much less than 1 A, so it'll be a waste of time driving the gate with "10V" if 6V would still do virtually the same. "Vgs=10V" is preferred of course it is, but 10V isn't available in this application, luckily, with some careful analysis with a low motor current, 6V would work, only just.
IRF510s are old school nowdays early generation devices. There are many lower Rds(on) and higher gm newer generation fets available that's better suited, that a 510 wouldn't be picked, so it's not really worth wasting effort going that deep into the analysis. But I will...
You can see from the IRF510's typical curve I posted, the 'typical' Rds(on) it has (@ Vgs=6V) = 0.6 ohm. 6V is the battery voltage. The effect of parametric spread on the shape of the 'typical curve' will take some imagination, based on worst case datasheet figures, but in practice in the real world it's actually much less when real samples are measured.
At the drain current of less than 1A, the fet is so well enhanced (Vgs=6V) into the 'ohmic conduction' part of the curve, that a fair bit of variation of actual Vgs(th) from 'typical' could be tolerated before the fet starts to drop out of ohmic conduction mode. In practice, Vgs(th) doesn't vary that much from typical, if it does more than 1V that's very bad luck, or the fet is an original 1980's one. Actual perfomance can be better than typical as well as worse than typical, and modern fets often out-perform the (old) datasheet.
What often happens with fets is that people don't understand the curves, temperature dependence and production spread variance properly. They'll get 'bitten' by a project that went wrong, then after that they mis-trust the datasheet blaming 'typical' data as 'mis-representative;, then as this advances, they cover themselves each time by interpreting all of the 'worst-case' data they can find and applying it all together at the same time (which is totally invalid of course, but it does minimise the risk of them ever making a mistake in their design).
Anyway, it's all academic really, because even though an IRF510 would work with 6V of gate drive, it would be better to use instead later generation automotive 1.5V Vgs(th) fets that have milliohm Rds(on), which are the same price.